Distance on a great circle

Average Error: 24.8 → 24.8

Time: 29.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

↓

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70369 = R;
        double r70370 = 2.0;
        double r70371 = phi1;
        double r70372 = phi2;
        double r70373 = r70371 - r70372;
        double r70374 = r70373 / r70370;
        double r70375 = sin(r70374);
        double r70376 = pow(r70375, r70370);
        double r70377 = cos(r70371);
        double r70378 = cos(r70372);
        double r70379 = r70377 * r70378;
        double r70380 = lambda1;
        double r70381 = lambda2;
        double r70382 = r70380 - r70381;
        double r70383 = r70382 / r70370;
        double r70384 = sin(r70383);
        double r70385 = r70379 * r70384;
        double r70386 = r70385 * r70384;
        double r70387 = r70376 + r70386;
        double r70388 = sqrt(r70387);
        double r70389 = 1.0;
        double r70390 = r70389 - r70387;
        double r70391 = sqrt(r70390);
        double r70392 = atan2(r70388, r70391);
        double r70393 = r70370 * r70392;
        double r70394 = r70369 * r70393;
        return r70394;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70395 = R;
        double r70396 = 2.0;
        double r70397 = phi1;
        double r70398 = phi2;
        double r70399 = r70397 - r70398;
        double r70400 = r70399 / r70396;
        double r70401 = sin(r70400);
        double r70402 = pow(r70401, r70396);
        double r70403 = cos(r70397);
        double r70404 = cos(r70398);
        double r70405 = r70403 * r70404;
        double r70406 = lambda1;
        double r70407 = lambda2;
        double r70408 = r70406 - r70407;
        double r70409 = r70408 / r70396;
        double r70410 = sin(r70409);
        double r70411 = r70405 * r70410;
        double r70412 = r70411 * r70410;
        double r70413 = r70402 + r70412;
        double r70414 = sqrt(r70413);
        double r70415 = 1.0;
        double r70416 = r70406 / r70396;
        double r70417 = sin(r70416);
        double r70418 = r70407 / r70396;
        double r70419 = cos(r70418);
        double r70420 = r70417 * r70419;
        double r70421 = cos(r70416);
        double r70422 = sin(r70418);
        double r70423 = r70421 * r70422;
        double r70424 = r70420 - r70423;
        double r70425 = cbrt(r70424);
        double r70426 = 6.0;
        double r70427 = pow(r70425, r70426);
        double r70428 = r70427 * r70410;
        double r70429 = cbrt(r70428);
        double r70430 = r70429 * r70429;
        double r70431 = cbrt(r70430);
        double r70432 = cbrt(r70429);
        double r70433 = r70431 * r70432;
        double r70434 = r70405 * r70433;
        double r70435 = r70434 * r70410;
        double r70436 = r70402 + r70435;
        double r70437 = r70415 - r70436;
        double r70438 = sqrt(r70437);
        double r70439 = atan2(r70414, r70438);
        double r70440 = r70396 * r70439;
        double r70441 = r70395 * r70440;
        return r70441;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 24.8

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
2. Using strategy rm

3. Applied add-cbrt-cube 24.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

4. Simplified 24.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{\color{blue}{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
5. Using strategy rm

6. Applied add-cube-cbrt 24.9

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}^{3}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

7. Simplified 24.9

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}^{6}}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}^{3}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
8. Using strategy rm

9. Applied div-sub 24.9

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}^{3}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

10. Applied sin-diff 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}^{3}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
11. Using strategy rm

12. Applied unpow3 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

13. Applied cbrt-prod 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sqrt[3]{\left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

14. Simplified 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6}} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

15. Simplified 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

16. Final simplification 24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}^{6} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

Reproduce

herbie shell --seed 2020046 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))